The log (short for logarithm) of a number N is the exponent used to raise a certain "base" number B to get N. In short, logB⁡N=x{\displaystyle \log _{B}\ N=x} means that Bx=N{\displaystyle B^{x}=N}.

Typically, logs use base 10. An increase of "1" in a base 10 log is equivalent to an increase by a power of 10 in normal notation. In logs, "3" is 100 times the size of "1". If the log is written without an explicit base, 10 is (usually) implied.

Another common base for logs is the trancendental number e{\displaystyle e}, which is approximately 2.7182818.... Since ddxex=ex{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}, these can be more convenient than log10{\displaystyle \log _{10}}. Often, the notation ln⁡x{\displaystyle \ln \ x} is used instead of loge⁡x{\displaystyle \log _{e}\ x}.

The following properties of logs are true regardless of whether the base is 10, e{\displaystyle e}, or some other number.

logA + logB = log(AB)

logA – logB = log(A/B)

log(AB) = B log(A)

Adding the log of A to the log of B will give the same result as taking the log of the product A times B.

Subtracting the log of B from the log of A will give the same result as taking the log of the quotient A divided by B.

The log of (A to the Bth power) is equal to the product (B times the log of A).